## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Hank', 'hours worked by Laura', 'hours worked by Mary', 'hours worked by Peggy']. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the natural language objects into symbolic notation
The symbolic representation of the variables is:
- $x_0$ : hours worked by Hank
- $x_1$ : hours worked by Laura
- $x_2$ : hours worked by Mary
- $x_3$ : hours worked by Peggy

## 3: Define the objective function in symbolic notation
The objective function to minimize is: $9.66x_0^2 + 2.25x_0x_1 + 5.78x_0x_2 + 9.13x_0x_3 + 7.01x_1^2 + 4.14x_2^2 + 3.29x_3^2 + 8.94x_2$.

## 4: List the constraints in symbolic notation
The constraints are:
- $12x_0 \leq 184$ (Hank's paperwork competence rating)
- $16x_0 \leq 229$ (Hank's likelihood to quit index)
- $11x_1 \leq 184$ (Laura's paperwork competence rating)
- $19x_1 \leq 229$ (Laura's likelihood to quit index)
- $7x_2 \leq 184$ (Mary's paperwork competence rating)
- $14x_2 \leq 229$ (Mary's likelihood to quit index)
- $4x_3 \leq 184$ (Peggy's paperwork competence rating)
- $13x_3 \leq 229$ (Peggy's likelihood to quit index)
- $12x_0 + 7x_2 \geq 32$
- $11x_1 + 7x_2 \geq 17$
- $11x_1 + 4x_3 \geq 21$
- $12x_0 + 11x_1 + 7x_2 \geq 40$
- $12x_0 + 11x_1 + 7x_2 + 4x_3 \geq 40$
- $16x_0 + 14x_2 \geq 47$
- $14^2x_2^2 + 13^2x_3^2 \geq 56$
- $19^2x_1^2 + 14^2x_2^2 \geq 22$
- $19x_1 + 13x_3 \geq 32$
- $16x_0 + 19x_1 \geq 36$
- $16x_0 + 13x_3 \geq 20$
- $16x_0 + 19x_1 + 14x_2 + 13x_3 \geq 20$
- $-2x_2 + 6x_3 \geq 0$
- $-8x_1^2 + 7x_2^2 \geq 0$
- $10x_0^2 - 8x_3^2 \geq 0$
- $11^2x_1^2 + 7^2x_2^2 \leq 184^2$
- $12x_0 + 4x_3 \leq 103$
- $12x_0 + 11x_1 \leq 52$
- $11x_1 + 4x_3 \leq 146$
- $12^2x_0^2 + 11^2x_1^2 + 4^2x_3^2 \leq 125^2$

## 5: Specify variable domains
- $x_0$ is a fractional number
- $x_1$ is an integer
- $x_2$ is an integer
- $x_3$ is a fractional number

## 6: Create the Gurobi model and variables
We will use Gurobi to solve this optimization problem.

```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by Hank
x1 = m.addVar(name="x1", lb=0, integrality=gp.GRB.Integer)  # hours worked by Laura
x2 = m.addVar(name="x2", lb=0, integrality=gp.GRB.Integer)  # hours worked by Mary
x3 = m.addVar(name="x3", lb=0)  # hours worked by Peggy

# Objective function
m.setObjective(9.66 * x0**2 + 2.25 * x0 * x1 + 5.78 * x0 * x2 + 9.13 * x0 * x3 + 
               7.01 * x1**2 + 4.14 * x2**2 + 3.29 * x3**2 + 8.94 * x2, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(12 * x0 <= 184)
m.addConstr(16 * x0 <= 229)
m.addConstr(11 * x1 <= 184)
m.addConstr(19 * x1 <= 229)
m.addConstr(7 * x2 <= 184)
m.addConstr(14 * x2 <= 229)
m.addConstr(4 * x3 <= 184)
m.addConstr(13 * x3 <= 229)

m.addConstr(12 * x0 + 7 * x2 >= 32)
m.addConstr(11 * x1 + 7 * x2 >= 17)
m.addConstr(11 * x1 + 4 * x3 >= 21)
m.addConstr(12 * x0 + 11 * x1 + 7 * x2 >= 40)
m.addConstr(12 * x0 + 11 * x1 + 7 * x2 + 4 * x3 >= 40)

m.addConstr(16 * x0 + 14 * x2 >= 47)
m.addConstr(14**2 * x2**2 + 13**2 * x3**2 >= 56)
m.addConstr(19**2 * x1**2 + 14**2 * x2**2 >= 22)
m.addConstr(19 * x1 + 13 * x3 >= 32)
m.addConstr(16 * x0 + 19 * x1 >= 36)
m.addConstr(16 * x0 + 13 * x3 >= 20)
m.addConstr(16 * x0 + 19 * x1 + 14 * x2 + 13 * x3 >= 20)

m.addConstr(-2 * x2 + 6 * x3 >= 0)
m.addConstr(-8 * x1**2 + 7 * x2**2 >= 0)
m.addConstr(10 * x0**2 - 8 * x3**2 >= 0)

m.addConstr(11**2 * x1**2 + 7**2 * x2**2 <= 184**2)
m.addConstr(12 * x0 + 4 * x3 <= 103)
m.addConstr(12 * x0 + 11 * x1 <= 52)
m.addConstr(11 * x1 + 4 * x3 <= 146)
m.addConstr(12**2 * x0**2 + 11**2 * x1**2 + 4**2 * x3**2 <= 125**2)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("x0: ", x0.varValue)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
    print("x3: ", x3.varValue)
else:
    print("The model is infeasible")
```

## Step 7: Symbolic Representation
Here is the symbolic representation of the problem:

```json
{
    'sym_variables': [
        ['x0', 'hours worked by Hank'], 
        ['x1', 'hours worked by Laura'], 
        ['x2', 'hours worked by Mary'], 
        ['x3', 'hours worked by Peggy']
    ], 
    'objective_function': '9.66*x0^2 + 2.25*x0*x1 + 5.78*x0*x2 + 9.13*x0*x3 + 7.01*x1^2 + 4.14*x2^2 + 3.29*x3^2 + 8.94*x2', 
    'constraints': [
        '12*x0 <= 184', 
        '16*x0 <= 229', 
        '11*x1 <= 184', 
        '19*x1 <= 229', 
        '7*x2 <= 184', 
        '14*x2 <= 229', 
        '4*x3 <= 184', 
        '13*x3 <= 229', 
        '12*x0 + 7*x2 >= 32', 
        '11*x1 + 7*x2 >= 17', 
        '11*x1 + 4*x3 >= 21', 
        '12*x0 + 11*x1 + 7*x2 >= 40', 
        '12*x0 + 11*x1 + 7*x2 + 4*x3 >= 40', 
        '16*x0 + 14*x2 >= 47', 
        '196*x2^2 + 169*x3^2 >= 56', 
        '361*x1^2 + 196*x2^2 >= 22', 
        '19*x1 + 13*x3 >= 32', 
        '16*x0 + 19*x1 >= 36', 
        '16*x0 + 13*x3 >= 20', 
        '16*x0 + 19*x1 + 14*x2 + 13*x3 >= 20', 
        '-2*x2 + 6*x3 >= 0', 
        '-8*x1^2 + 7*x2^2 >= 0', 
        '10*x0^2 - 8*x3^2 >= 0', 
        '121*x1^2 + 49*x2^2 <= 33856', 
        '12*x0 + 4*x3 <= 103', 
        '12*x0 + 11*x1 <= 52', 
        '11*x1 + 4*x3 <= 146', 
        '144*x0^2 + 121*x1^2 + 16*x3^2 <= 15625'
    ]
}
```